Bend & Snap: Origami Inspires New Ways to Fold Curved Objects

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A new mathematical rule explains how simple, 3D curved surfaces —
such as domes or saddles — can be folded and snapped into new
positions or to form different structures.

Typically, snapping metal in half isn't a useful operation, but
some objects could benefit from such
innovative folding techniques. For instance, parts of a
satellite need to collapse down for storage during launch but
then quickly expand in space. Future robots could be
more practical if they are able to reconfigure their arms without
the need of moving parts. As such, understanding how to bend
materials smoothly or snap them quickly could enable more
efficient mechanical designs, said Arthur Evans, a postdoctoral
researcher in the Department of Mathematics at the University of
Wisconsin-Madison.

Origami artists usually fold flat sheets of paper to create
shapes or structures. But folding materials with curves (such as
dome- or saddle-shaped objects) usually means the finished
product will be stiffer and stronger. This is similar to how
folding a flat pizza slice into a cylinder-type shape helps keep
the slice rigid.

The
Venus flytrap is a domelike plant with leaves that are shaped
like clamshells. When a fly brushes past the plant's sensitive
hairs, it quickly folds the dome back together, snapping shut
(like a spring mechanism without springs).

Engineers have used this snapping technique to build satellite
airfoils that can collapse and expand, and to design tiny
spherical particles that lock together. But researchers don't yet
have theories to explain when or why it happens, Evans said.

Robert Lang, a physicist-turned-origami artist, published one of
the earliest studies on folding nonflat surfaces in the journal
The Mathematical Intelligencer in 2012. The research showed
how to take paper curved in the 3D shape of a saddle (akin to a
Pringles chip) and fold it into a crane.

In their new study, Evans and his colleagues found a general
mathematical rule that explains whether a curved surface will
either snap or bend smoothly when folded. The rule takes into
account only the geometric shape of an object, not its material
or size.

To understand the mathematical rule, imagine a cylinder and a
straight piece of wire. If the wire can wrap along the cylinder
and doesn't deform it in any way, then you can fold the cylinder
along that curve without snapping it.

If instead the wire bends tightly around the cylinder so that it
strains to straighten out, then it will pull the cylinder and
expand it slightly. If a curve pulls on any curved surface like
this, the curve will snap when folded.

"The equations cover [folding] any kind of surface you could
possibly think of," Evans said.

To experimentally test this rule, the team looked at three
so-called shell shapes that mathematically represent all the
different cases of curvature: the cylinder, the sphere and the
spiral-staircase-shaped helicoid. The researchers found that, in
general, the sphere always snaps while the helicoid bends along
two special paths and snaps everywhere else.

Evans and his colleagues created 3D-printed
models made out of dental rubber and plastic and
strategically poked the models to examine how they deformed from
different forces at different distances.

The researchers have not yet demonstrated any applications for
the theory, but since the rule depends only on the shape of the
surface, it can be applied to any material of any size, they
said.

For example, at the microscopic scale, Evans speculated that
knowing which curves snap quickly could someday help researchers
create tiny snapping cells or capsules that could mix together
liquids, such as drugs going into the human body, faster than
mixing methods available today.

"They put together an elegant theory," Ashkan Vaziri, an
engineering researcher at Northeastern University in Boston, who
was not involved in the study but has studied such shapes, told
Live Science.

Now, Evans and his colleagues said they are thinking about how to
use their findings to design structures that can collapse down
and lock into place, such as new, collapsible satellite airfoils.
Engineers have been making locking structures that take advantage
of bending or snapping for a while, but knowing a rule for such
structures before they are designed would be more efficient,
Evans said. Engineers could then pre-crease any curved object in
just the right spots so that when it's pushed or slapped, it
snaps or slowly bends into a different, predesigned
configuration.

But for now, researchers only know for sure what happens to a
single fold.

One of the next steps might be to investigate how to connect
multiple folds together to create more-complicated structures,
the researchers said. In the future, scientists might also
investigate how to get structures to automatically bend or snap
without being pushed or slapped.